Validation 3. Laminar Flow Around a Circular Cylinder 3.1 Introduction Steady and unsteady laminar flow behind a circular cylinder, representing flow around bluff bodies, has been subjected to numerous experimental and computational studies. Most of the experimental studies are based on flow visualization and they indicate that as the Reynolds number Re increases, the flow shows a series of different structures [1]. One of the most prominent flow structure changes takes place in the vicinity of Re = 40. Below this Reynolds number, the flow is steady and characterized by the presence of a symmetric pair of closed separation bubbles. Beyond Re = 40, the flow becomes unsteady and asymmetric, and alternate vortex shedding begins. 3.2 Purpose The purpose of this test is to validate FLUENT s ability to predict the flow structure as well as the reattachment length and Strouhal number against experimental results. The present calculations were confined to the low-reynolds-number regime (Re = 20, Re = 40, and Re = 100), which encompasses steady symmetrical separated flow as well as unsteady asymmetric flow. The results were compared with the flow visualizations presented in [1] and the experimental data in [2]. 3.3 Problem Description An infinitely long circular cylinder of diameter D = 2.0 m is placed in an otherwise undisturbed uniform crossflow (U = 1.0 m/s) as shown in Figure 3.3.1. The lateral boundary and the exit boundary in the far wake are placed at 5D and 20D from the center of the circular cylinder, respectively. 3.3.1 Fluid Properties The properties of the fluid are assumed to be constant, as shown in Table 3.3.1. 3.3.2 Flow Physics The Reynolds number is based on the cylinder diameter and the free stream velocity. Different values of Re (20, 40, 100) in the simulation were obtained by changing the viscosity. c Fluent Inc. September 13, 2002 3-1
Laminar Flow Around a Circular Cylinder u=1.0, v=0.0 Flow U = 1.0 T = 273 K D u=0.0 v=0.0 T = 373 K W 10D 20D Figure 3.3.1: Problem Description Table 3.3.1: Fluid Properties Reynolds No., Re 20 40 100 Density, ρ 1 kg/m 3 1 kg/m 3 1 kg/m 3 Viscosity, µ 0.1 Pa-s 0.05 Pa-s 0.02 Pa-s Re = ρud µ (3.3-1) The experimental Strouhal number for the Re = 100 case is 0.165, derived from [2] and defined as S = D τu (3.3-2) where τ is the period of the vortex shedding. 3.3.3 Boundary Conditions The no-slip wall condition is applied to the cylinder wall. Uniform free stream conditions (U = 1.0 m/s) are applied at the inlet and lateral boundaries. The flow exit is treated as a zero-normal-gradient outlet boundary. 3.4 Grid A51 51 quadrilateral mesh was used in this validation study, and is shown in Figure 3.4.1. 3-2 c Fluent Inc. September 13, 2002
3.5 Case Setup velocity-inlet-3 velocity-inlet-7 outflow velocity-inlet-3 Figure 3.4.1: Grid 3.5 Case Setup The FLUENT case was set up using constant fluid properties and the boundary conditions described in Section 3.3. Three runs were made using different values of viscosity to yield three different Reynolds numbers (Re = 20, 40, and 100) as shown in Table 3.5.1. Table 3.5.1: Run Conditions Run 1 2 3 Reynolds Number, Re 20 40 100 3.6 Calculation The second-order discretization scheme was used throughout this study. The first two cases (Re = 20, 40) were run for steady-state solutions and the third case (Re = 100) was modeled as transient with a time step of 0.2 s. For velocity-pressure coupling, the SIMPLEC algorithm was used. When the SIMPLEC algorithm is used, the pressure under-relaxation is automatically increased to 1.0. (The setting for pressure under-relaxation is 0.3 when the default SIMPLE algorithm is used). Here the momentum under-relaxation has also been increased from 0.7 to 0.9 to yield better convergence. To break the symmetry of the flow and eventually trigger the vortex shedding, artificial perturbation was applied. The flow was perturbed by patching a uniform x velocity of 1 m/s in the upper half of the domain and 0 m/s in the lower half. This is done using a custom field function defined as u = y + y 2y (3.6-1) c Fluent Inc. September 13, 2002 3-3
Laminar Flow Around a Circular Cylinder where u is the x velocity. This custom field function is saved in the case file with the name initialvelocity. After initializing the flow, the x velocity of the entire fluid zone must be patched with this custom field function. It should be noted that the use of artificial perturbation is not mandatory. It was applied here to expedite the symmetry breaking and subsequent vortex shedding. 3.7 Results 3.7.1 Re =20and Re =40 The near-wake contours of stream function and the velocity vectors for the Re =20and Re = 40 cases are shown in Figures 3.7.1 3.7.4. The flows are seen to be practically symmetric. The reattachment length is measured from the downstream side of the cylinder to the point where the x velocity changes sign from negative to positive. In this FLUENT model, a line named centerline and aligned with the x axis was created. XY plots of x velocity along the centerline are shown in Figures 3.7.5 and 3.7.6. The reattachment lengths, normalized by the cylinder radius (R = 1 m), are shown in Table 3.7.1. Figure 3.7.7, which was taken from [2], shows the time evolution of the reattachment length collected from various sources. The present results are seen to agree fairly well with the data from [2]. Table 3.7.1: FLUENT Dimensionless Reattachment Lengths (L A ) Re =20 Re =40 L A 1.85 4.27 3.7.2 Re = 100 Figures 3.7.8 3.7.17 show the instantaneous velocity vector field and the corresponding streamlines at the five phases during one cycle of the vortex shedding. The alternate formation, convection, and diffusion of the vortices are clearly seen. To quantify the periodicity of the flow, the time history of the y velocity at a point situated 1 m behind the cylinder in the near wake (x, y) =(2, 0) was recorded and is shown in Figure 3.7.18. The average period was found to be τ =12.1 s, which is equivalent to a Strouhal number of 0.165. Figure 3.7.19, taken from [2], shows a collection of data from many sources on the Strouhal number vs. Reynolds number relationship. The FLUENT result agrees well with an average of the data shown in the figure. 3-4 c Fluent Inc. September 13, 2002
3.7 Results 9.99e+00 9.98e+00 9.98e+00 9.97e+00 9.96e+00 9.96e+00 9.95e+00 Laminar Flow Over a Cylinder ( Re=20) Contours of Stream Function (kg/s) FLUENT 6.0 (2d, segregated, lam) Figure 3.7.1: Stream Function Contours in the Wake (Re = 20) 9.99e+00 9.98e+00 9.98e+00 9.97e+00 9.96e+00 9.96e+00 9.95e+00 Laminar Flow Over a Cylinder ( Re=40) Contours of Stream Function (kg/s) FLUENT 6.0 (2d, segregated, lam) Figure 3.7.2: Stream Function Contours in the Wake (Re = 40) c Fluent Inc. September 13, 2002 3-5
Laminar Flow Around a Circular Cylinder 6.00e-01 5.60e-01 5.20e-01 4.80e-01 4.40e-01 4.00e-01 3.61e-01 3.21e-01 2.81e-01 2.41e-01 2.01e-01 1.61e-01 1.21e-01 8.11e-02 4.12e-02 1.30e-03 Laminar Flow Over a Cylinder ( Re=20) Velocity Vectors Colored By Velocity Magnitude (m/s) FLUENT 6.0 (2d, segregated, lam) Figure 3.7.3: Velocity Vectors in the Wake (Re = 20) 7.85e-01 7.33e-01 6.81e-01 6.28e-01 5.76e-01 5.24e-01 4.72e-01 4.20e-01 3.67e-01 3.15e-01 2.63e-01 2.11e-01 1.59e-01 1.06e-01 5.43e-02 2.09e-03 Laminar Flow Over a Cylinder ( Re=40) Velocity Vectors Colored By Velocity Magnitude (m/s) FLUENT 6.0 (2d, segregated, lam) Figure 3.7.4: Velocity Vectors in the Wake (Re = 40) 3-6 c Fluent Inc. September 13, 2002
3.7 Results centerline 5.00e-03 0.00e+00-5.00e-03-1.00e-02 Velocity X (m/s) -1.50e-02-2.00e-02-2.50e-02-3.00e-02-3.50e-02-4.00e-02 0 0.5 1 1.5 2 Position (m) 2.5 3 Laminar Flow Over a Cylinder ( Re=20) X Velocity FLUENT 6.0 (2d, segregated, lam) Figure 3.7.5: x Velocity Along the Centerline (Re = 20) centerline Velocity X (m/s) 4.00e-02 2.00e-02 0.00e+00-2.00e-02-4.00e-02-6.00e-02-8.00e-02-1.00e-01-1.20e-01 0 1 2 3 4 Position (m) 5 6 Laminar Flow Over a Cylinder ( Re=40) X Velocity FLUENT 6.0 (2d, segregated, lam) Figure 3.7.6: x Velocity Along the Centerline (Re = 40) c Fluent Inc. September 13, 2002 3-7
Laminar Flow Around a Circular Cylinder Figure 3.7.7: Reattachment Length vs. Time (from [2]) 3-8 c Fluent Inc. September 13, 2002
3.7 Results 9.98e+00 9.95e+00 9.93e+00 9.90e+00 9.88e+00 9.85e+00 Contours of Stream Function (kg/s) (Time=4.2200e+01) Jun 06, 2002 Figure 3.7.8: Stream Function Contours in the Wake (Re = 100, t =42.2 s) 8.00e-01 7.47e-01 6.94e-01 6.41e-01 5.88e-01 5.35e-01 4.82e-01 4.28e-01 3.75e-01 3.22e-01 2.69e-01 2.16e-01 1.63e-01 1.10e-01 5.69e-02 3.81e-03 Velocity Vectors Colored By Velocity Magnitude (m/s) (Time=4.2200e+01) Figure 3.7.9: Velocity Vectors in the Wake (Re = 100, t =42.2 s) c Fluent Inc. September 13, 2002 3-9
Laminar Flow Around a Circular Cylinder 9.99e+00 9.97e+00 9.95e+00 9.93e+00 9.91e+00 9.89e+00 9.87e+00 9.85e+00 Contours of Stream Function (kg/s) (Time=4.5200e+01) Figure 3.7.10: Stream Function Contours in the Wake (Re = 100, t =45.2 s) 8.00e-01 7.47e-01 6.94e-01 6.41e-01 5.88e-01 5.35e-01 4.82e-01 4.28e-01 3.75e-01 3.22e-01 2.69e-01 2.16e-01 1.63e-01 1.10e-01 5.69e-02 3.81e-03 Velocity Vectors Colored By Velocity Magnitude (m/s) (Time=4.5200e+01) Figure 3.7.11: Velocity Vectors in the Wake (Re = 100, t =45.2 s) 3-10 c Fluent Inc. September 13, 2002
3.7 Results 9.99e+00 9.97e+00 9.95e+00 9.93e+00 9.91e+00 9.89e+00 9.87e+00 9.85e+00 Contours of Stream Function (kg/s) (Time=4.8200e+01) Figure 3.7.12: Stream Function Contours in the Wake (Re = 100, t =48.2 s) 8.00e-01 7.47e-01 6.94e-01 6.41e-01 5.88e-01 5.35e-01 4.82e-01 4.28e-01 3.75e-01 3.22e-01 2.69e-01 2.16e-01 1.63e-01 1.10e-01 5.69e-02 3.81e-03 Velocity Vectors Colored By Velocity Magnitude (m/s) (Time=4.8200e+01) Figure 3.7.13: Velocity Vectors in the Wake (Re = 100, t =48.2 s) c Fluent Inc. September 13, 2002 3-11
Laminar Flow Around a Circular Cylinder 9.99e+00 9.97e+00 9.95e+00 9.93e+00 9.91e+00 9.89e+00 9.87e+00 9.85e+00 Contours of Stream Function (kg/s) (Time=5.1200e+01) Figure 3.7.14: Stream Function Contours in the Wake (Re = 100, t =51.2 s) 8.00e-01 7.47e-01 6.94e-01 6.41e-01 5.88e-01 5.35e-01 4.82e-01 4.28e-01 3.75e-01 3.22e-01 2.69e-01 2.16e-01 1.63e-01 1.10e-01 5.69e-02 3.81e-03 Velocity Vectors Colored By Velocity Magnitude (m/s) (Time=5.1200e+01) Figure 3.7.15: Velocity Vectors in the Wake (Re = 100, t =51.2 s) 3-12 c Fluent Inc. September 13, 2002
3.7 Results 9.99e+00 9.97e+00 9.95e+00 9.93e+00 9.91e+00 9.89e+00 9.87e+00 9.85e+00 Contours of Stream Function (kg/s) (Time=5.4600e+01) Figure 3.7.16: Stream Function Contours in the Wake (Re = 100, t =54.6 s) 8.00e-01 7.47e-01 6.94e-01 6.41e-01 5.88e-01 5.35e-01 4.82e-01 4.28e-01 3.75e-01 3.22e-01 2.69e-01 2.16e-01 1.63e-01 1.10e-01 5.69e-02 3.81e-03 Velocity Vectors Colored By Velocity Magnitude (m/s) (Time=5.4600e+01) Figure 3.7.17: Velocity Vectors in the Wake (Re = 100, t =54.6 s) c Fluent Inc. September 13, 2002 3-13
Laminar Flow Around a Circular Cylinder Average Y Velocity 7.00e-01 6.00e-01 5.00e-01 4.00e-01 3.00e-01 Velocity Magnitude (m/s) 2.00e-01 1.00e-01 0.00e+00-1.00e-01-2.00e-01-3.00e-01-4.00e-01 0 5 10 15 20 25 Time (s) 30 35 40 45 Velocity Magnitude (Time=4.2200e+01) Feb 06, 2002 Figure 3.7.18: y Velocity History (Re = 100, t =42.2 s) 3-14 c Fluent Inc. September 13, 2002
3.8 Conclusion Figure 3.7.19: Strouhal Number vs. Reynolds Number (from [2]) 3.8 Conclusion FLUENT has been validated for a classical example of external flows around bluff bodies. The results agree fairly well with the data from various sources, in terms of the length of the recirculation region in the two steady cases (Re = 20, Re = 40), and the vortex shedding frequency in the unsteady case (Re = 100). 3.9 References 1. Coutanceau, M. and Defaye, J.R., Circular Cylinder Wake Configurations A Flow Visualization Survey, Appl. Mech. Rev., 44(6), June 1991. 2. Braza, M., Chassaing, P., and Minh, H.H., Numerical Study and Physical Analysis of the Pressure and Velocity Fields in the Near Wake of a Circular Cylinder, J. Fluid Mech., 165:79 130, 1986. c Fluent Inc. September 13, 2002 3-15